Linear Algebra Study Guide for Exam 3

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Oct 10, 2013 (5 years and 8 months ago)

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Linear Algebra

Study Guide for Exam
3

Exam 3 is on Chapter 4. We covered 4.1 thru 4.7.

I will NOT give you the theorems on this exam. You need to know them.
Most

were introduced in chapter 2 and should be very familiar to you.

You may use a graphing c
alculator. The only types I will not allow are those that
do symbolic manipulations (e.g., TI
-
89).

You will get a take
-
home part of the exam.

Section
4.1

You need to know the definitions for or how to do
/use

the following:

Prove or disprove a given set
is a vector space. This will be on the take home.

Check that a given set is a subspace of a vector space.

The span of a set of vectors forms a subspace.

Section
4
.2

You need to know the definitions for or how to do
/use

the following:

Definition of
the nu
ll space.

The null space forms a subspace of R
n
.

Given a matrix A, find the null space explicitly.

Definition of the column space.

The column space of A is a subspace of R
m
.

Know the fact in the blue box on page 230.

The definition for a linear transformat
ion on page 232.

Know the definition for the kernel and range of a linear transformation. This is
just below the green box on page 232 or get it from your lecture notes.

Section
4
.3

You need to know the definitions for or how to do
/use

the following:

The

definition of a linearly independent set which is on page 237 (equation 1).

Know theorem 4.

Definition of a basis for a subspace of a vector space (also is the definition for the
basis of the vector space itself). See page 238.

The Spanning Set Theorem o
n page 239.

How to find a basis for Nul(A) and Col(A).

Read “Two Views of a Basis” on page 242.

Section
4
.4

You need to know the definitions for or how to do/use the following:

Know the content of the Unique Representation Theorem (page 246). The
coordin
ates of a vector with respect to a given basis are unique.

The definition on page 246; just be familiar with the notation.

Know how to find the change of coordinates matrix. See equation 4 on page 249.

Theorem 8.

Section
4
.5

You need to know the defin
itions for or how to do
/use

the following:

Be clear on the content of theorems 9 and 10.

Know the definition of dimension on page 257.

Know theorems 11 and 12.

Section
4
.6

You need to know the definitions for or how to do/use the following:

Know the defin
ition for the row space of A. See page 263 or your notes.

Theorem 13.

Know how to find a basis for
Row (
A) given a matrix A.

Know the definition of rank; page 265.

The Rank Theorem.

The new entries in the Invertible Matrix Theorem on page 267.

Section
4
.
7

You need to know the definitions for or how to do/use the following:

Know how to find the coordinates of a given vector with respect to different
bases.

Theorem 15.

Section

4.8 and 4
.9

We skipped 4.8 and 4.9.

To study for the exam, know the above list

of topics. Do your homework. Review your
quizzes. Do the supplementary exercises on pp.
298
-
300

(odds are in the back of the
book).